Let $X_1, X_2, X_3, X_4$ be i.i.d continuous random variables with a common distribution function $F$. How to prove that "all the 4! possible orderings of $X_1, \dots, X_4$ are equally likely" without calculating probability of any ordering by integrating the joint density function?
I guess the "exchangablity" property of i.i.d random variable needs to be used. But this property is proved by integrating the joint density function.
Another thing I also want to know the geometric intuitive solution of the problem.